Meaningfulness and Invariance

Few disavow the principle that scientific propositions should be meaningful in the sense of asserting something that is verifiable or falsifiable about the qualitative or empirical situation under discussion. What makes this principle tricky to apply in practice is that much of what is said is formulated not as simple assertions about qualitative or empirical events – such as a certain object sinks when placed in water – but as laws formulated in rather abstract, often mathematical, terms. It is not always apparent exactly what class of qualitative observations corresponds to such (often numerical) laws. Theories of meaningfulness are methods for investigating such matters, and invariance concepts are their primary tools. The problem of meaningfulness, which has been around since the inception of mathematical science in ancient times, has proved to be difficult and subtle; even today it has not been fully resolved. This entry surveys some of the current ideas about it, and illustrates, through examples, some of its uses. The presentation requires some elementary technical concepts of measurement theory (such as representation, scale type, etc.), which are explained in Measurement, Theory of.